Framing Competition∗ Michele Piccione† and Ran Spiegler‡ May 4, 2009

Abstract We analyze a model of market competition in which two identical firms choose prices as well as how to present, or “frame”, their products. A consumer is randomly assigned to one firm, and whether he makes a price comparison with the other firm is a probabilistic function of the firms’ framing strategies. We analyze Nash equilibria in this model. In particular, we show how the answers to the following questions are linked: (1) Are firms’ choices of prices and frames correlated? (2) Can firms earn payoffs in excess of the max-min level? (3) Does greater consumer rationality (in the sense of better ability to make price comparisons) imply lower equilibrium prices? We also argue that our model provides a novel account of the phenomenon of product differentiation.

1

Introduction

Standard models of market competition assume that consumers are perfectly able to form a preference ranking of all the alternatives they are aware of, given search costs and potentially limited information about product characteristics. In reality, consumers do not always carry out all the comparisons that “should” be made. Moreover, whether consumers make preference comparisons often depends on the way the alternatives are presented, or “framed”. For instance: ∗

We thank Noga Alon, Eddie Dekel, Kfir Eliaz, Sergiu Hart, Emir Kamenica, Ariel Rubinstein, Jakub Steiner, Jonathan Weinstein and seminar participants at Harvard/MIT, Penn, Princeton, and Tel Aviv. Spiegler acknowledges financial support from the European Research Council, Grant no. 230251, as well as the ESRC (UK). † London School of Economics. E-mail: [email protected]. ‡ University College London. URL: http://www.homepages.ucl.ac.uk/~uctprsp. E-mail: [email protected].

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• The complexity of prices often makes it hard to make comparisons. Prices can be defined for different units of measurement that a consumer may find difficult to convert to a common standard. Price schedules in industries such as communication, energy, and insurance often condition on a large number of partially overlapping contingencies, and such overlaps can hinder comparisons. • Consumers may fail to regard a market alternative as being relevant to their choice problem, even when they know of its existence. Few conspicuous features of the first alternative they consider may steer them towards making comparisons with some products at the expense of others. For instance, a consumer who is exposed to a hamburger ad or walks by a hamburger stall while considering options for a light meal, may fail to take into account alternatives that do not easily fall into the fast food category with which hamburger is traditionally associated.1

This paper studies market competition when consumers have limited ability to compare market alternatives, and when comparability is sensitive to framing. Adapting a formalism first introduced in Eliaz and Spiegler (2007), we construct a model that enriches standard Bertrand competition by incorporating the firms’ framing decisions. We are interested in the effects of framing on consumer behavior only in so far as it hinders or facilitates price comparisons, and we ignore framing effects that cause preference reversals. We explore the interaction between firms’ pricing and framing decisions, and its implications for industry profits and consumer welfare. Here are some of the questions that we address: (1) Are pricing and framing equilibrium strategies correlated? (2) Does the consumers’ limited, frame-sensitive ability to rank alternatives enable firms to earn collusive profits? (3) How are firms’ equilibrium pricing and framing decisions affected when some ways of framing an alternative are more conducive to price comparisons than others? (4) Does greater consumer rationality (in the sense of lower sensitivity to framing) lead to a more competitive equilibrium outcome? In our model, two profit-maximizing firms produce perfect substitutes at zero cost, and face one consumer who buys one unit if priced below a reservation value. Each firm i choose a price pi and a format xi for its products. Given the firms’ pricing and framing decisions, the consumer chooses as follows. He is initially assigned to one firm at random, say firm 1. With probability π (x1 , x2 ), the consumer makes a price comparison and chooses the rival firm’s product if strictly cheaper. Otherwise, he buys 1

This example is based on an experiment by Nedungadi (1989).

2

from the firm 1. When π (x, y) = π (y, x) for all formats x, y - a property we dub “order independence” - price comparisons are independent of the order in which the consumer considers alternatives. The framing structure given by π can be viewed as a random graph, where the set of nodes corresponds to the set of formats, and π (x, y) is the (independent) probability of a directed link from node x to node y. The graph structure represents the consumer’s limited, frame-sensitive ability to make price comparisons. We propose two interpretations for a link from format x to format y: (i) y is easy to compare to x; (ii) x triggers associations that make the consumer think of the product framed by y as a relevant choice whenever he first considers the product framed by x. Because of the graph structure, our framework may be reminiscent of models of spatial competition. However, in the concluding section we will illustrate significant differences between the two formalisms, both at the level of individual consumer behavior and at the level of equilibrium analysis. Formats in our model capture the various ways in which firms can present an intrinsically homogeneous product. We use the term “format” in a broad sense that includes aspects of the products’ presentation which may be of no relevance to a consumer’s utility and yet affect his propensity to make a price comparison. A format can be a price format, a “language” in which a contract is written, an aspect of the positioning of a product (e.g., the assignment of food products into categories such as snacks or health food), and so on. The utility-irrelevance of framing is a limitation of our model. For instance, a consumer may have preferences over the different contingencies covered by a mobile phone calling plan whereas, in our model, such contingencies are introduced by firms for the sole purpose of facilitating or hindering price comparisons. The benchmark case of a rational consumer is represented by a complete graph (i.e., every node is linked to every other node with probability one), because the consumer always makes a price comparison and chooses the cheapest alternative. The model collapses to conventional Bertrand competition, with firms charging prices equal to zero in Nash equilibrium.

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An illustrative example: A “core-periphery” graph We use the following example to illustrate the model and some of our main insights. Consider the order-independent graph given by Figure 1:2

q

Figure 1

The two “core” nodes in the center can be interpreted as relatively basic price formats that are comparable (and thus linked) with probability q. The four “peripheral” nodes represent more complex formats, each being comparable to one of the basic formats, to which it is linked with probability one. Alternatively, the core nodes may represent broad product categories, while each peripheral node can be interpreted as a refinement of its “parent” broad category. Let us consider first an extreme case in which the two core formats are incomparable - i.e., q = 0. The game played between the two firms has a unique symmetric Nash equilibrium. Firms play a mixed strategy that randomizes independently over formats and prices. The framing strategy assigns probability 12 to each of the two core formats and zero probability to the peripheral formats. Note that this framing strategy has the property that when a firm adopts it, the probability of a price comparison is 12 , independently of the rival firm’s framing strategy: it max-minimizes the probability of a price comparison. The expected equilibrium price is 12 , and thus firms earn an equilibrium payoff of 14 , which is also the max-min payoff. Profits are positive due to consumers’ limited ability to make price comparisons. However, competitive forces are strong enough to rule out additional, collusive gains above max-min payoffs. 2

In this paper, diagrams that represent order-independent graphs are drawn as non-directed graphs. In addition, the diagrams supress self-links. Order-independent graphs and non-directed graphs are payoff equivalent for the firms. The difference is that in the former the link between x and y is realized independently of the link between y and x whereas in the latter they are realized simultaneously.

4

Now consider the case in which the two core formats are comparable - i.e., q = 1. The framing strategy that mixes uniformly between the two core formats remains the unique strategy that max-minimizes the probability of a price comparison. Consequently, the max-min payoff is still 14 . However, in symmetric Nash equilibrium firms do not play this framing strategy. Instead, they mix uniformly over all six formats. Moreover, the framing and pricing strategies are correlated: when the price is in [ 23 , 1], firms mix uniformly over the four peripheral formats, and when the price is in [ 25 , 23 ], firms mix uniformly over the two core formats. Expected equilibrium price is 23 , and thus the firms’ equilibrium payoff is 13 , which exceeds the max-min level. The graph with q = 1 has greater connectivity than the graph with q = 0, and thus represents a “more rational” consumer. For any strategy profile of the firms, it leads to fewer decision errors for a consumer. Nevertheless, the expected Nash equilibrium price is higher when q = 1. This apparent anomaly is explained by the fact that firms that charge a high (low) price have an incentive to adopt a framing strategy that induces a low (high) probability of a price comparison. When q = 0, the framing strategy that mixes uniformly over the two core formats equalizes the probability of a price comparison for all formats. Hence, it is optimal for firms to adopt this framing strategy independently of their price. In contrast, when q = 1, mixing uniformly over the two core formats does not suit firms that charge a high price, as core formats are always comparable. When a firm’s realized price is high, it is optimal for the firm to choose peripheral formats as are they are less likely to trigger a price comparison. Thus, equilibrium payoffs rise above the max-min level. Overview of the results We begin our analysis of Nash equilibria for graphs that satisfy order independence. This analysis, presented in Section 3, highlights a property of graphs, called “weighted regularity”, which generalizes the familiar regularity property. A graph is weightedregular if nodes can be assigned weights such that each node has the same total weighted links. (Regularity corresponds to a special case in which the weights are uniform across the entire set of graph nodes.) Under weighted regularity, all formats are equally comparable, once the frequency with which they are used is factored in. We show that if a graph is weighted-regular, there exists a Nash equilibrium in which the firms’ pricing and framing strategies are independent, and their payoffs are equal to the max-min level. The significance of max-min equilibrium payoffs is that competitive forces prevail in that they push industry profits to the lowest level possible given the consumer’s limited ability to make price comparisons. Conversely, if firms’ pricing and framing strategies are independent in some Nash equilibrium, the 5

graph must be weighted-regular and firms earn max-min payoffs in this equilibrium. Moreover, their pricing strategies must be identical. We investigate a special class of symmetric Nash equilibria, called “cutoff equilibria”, where every format that is played with positive probability is unambiguously associated with prices either above or below a cutoff. We show that a cutoff equilibrium induces max-min payoffs if and only if the graph is weighted-regular. Moreover, the equilibrium framing strategy conditional on prices above (below) the cutoff minmaximizes (max-minimizes) the probability of a price comparison. We apply the results above to obtain a complete characterization of symmetric Nash equilibria in a class of “bi-symmetric” graphs, that is, graphs in which the connectivity between two formats depends only on which of two categories they belong to. In Section 4, we relax order independence and examine the extent to which these results can be extended. Related literature Our paper joins recent attempts to formalize in broad terms the role of framing effects in decision making. Rubinstein and Salant (2008) study choice behavior, where the notion of a choice problem is extended to include both the choice set A and a frame f , which is interpreted as observable information which should not affect the rational assessment of alternatives but nonetheless affects choice. A choice function assigns an element in A to every extended choice problem. Rubinstein and Salant conduct a choice-theoretic analysis of such extended choice functions, and relate their framework to the standard model of choice correspondences. In particular, they identify conditions under which extended choice functions are consistent with utility maximization. Bernheim and Rangel (2007) use a similar framework to extend standard welfare analysis to situations in which choices are sensitive to frames. Our notion of frame dependence differs from the one in the above models. First, we associate frames with individual alternatives, rather than entire choice sets. Second, in our model framing affects the probability that consumers apply a preference ranking, but never leads to preference reversals. Finally, our focus is on market implications rather than choice-theoretic analysis. In this respect, this paper is closest to Eliaz and Spiegler (2007), which first formalized the idea that framing (and marketing devices in general) affects preference incompleteness without reversing preference rankings. The model of consumer behavior in Eliaz and Spiegler is more general in that the consumer’s propensity to apply a preference ranking to a pair of market alternatives depends on the alternatives’ payoff-relevant details as well as their frames. In the market applications analyzed in Eliaz and Spiegler, framing decisions are costly and price setting is assumed 6

away, leading to quite different game-theoretic analysis. This paper contributes to a growing theoretical literature on the market interaction between profit-maximizing firms and boundedly rational consumers. Rubinstein (1993) analyzes monopolistic behavior when consumers differ in their ability to understand complex pricing schedules. Piccione and Rubinstein (2003) study intertemporal pricing when consumers have diverse ability to perceive temporal patterns. Spiegler (2006a,b) analyzes markets in which profit-maximizing firms compete over consumers who rely on naive sampling to evaluate each firm. Gabaix and Laibson (2006) and Eliaz and Spiegler (2008) study interaction with consumers having limited awareness of future contingencies. See Ellison (2006) for a recent survey. There are a few points of contact between our paper and the more traditional Industrial Organization literature. Varian (1980) was among the first to analyze price competition when some consumers fail to make price comparisons (these “loyal” consumers are not informed of any firm except the one they are assigned to). In Varian’s model, the fraction of “loyal” consumers is exogenous, whereas in our model it is a consequence of the firms’ framing decisions. Our paper is also related to the large literature on product differentiation (for instance, see Anderson, De Palma and Thisse (1992)). Indeed, our model provides a novel interpretation of this phenomenon. In equilibrium, firms offer a homogenous product in a variety of guises, and this variety can be viewed as a kind of product differentiation. Yet, in our model, differentiation does not result from the firms’ attempt to cater to diverse taste niches, but from the attempt to make price comparison less likely. The force behind differentiation is the limited comparability between different ways of presenting a homogeneous product, rather than differentiated tastes. Consequently, differentiation in our model has a purely negative effect on consumer welfare.

2

The Model

A graph is a pair (X, π), where X is a finite set of nodes and π : X × X → [0, 1] is a function that determines the probability π(x, y) with which a directed edge links node x to node y. The probability that x is linked to y is independent of other links being realized. Let n denote |X|. We refer to nodes as formats. Assume that π(x, x) = 1 for every x ∈ X - that is, every format is linked to itself. A graph π is deterministic if for every distinct x, y ∈ X, π(x, y) ∈ {0, 1}. A graph π is order independent if π(x, y) = π(y, x) for all x, y ∈ X. A market consists of two identical, expected-profit maximizing firms and one con7

sumer. These firms produce at zero cost a homogenous product for which the consumer has a reservation value equal to one. The firms move simultaneously. A pure strategy for firm i is a pair (pi , xi ), where pi ∈ [0, 1] is a price and xi ∈ X is a format. We allow ¢ ¡ firm i to employ mixed strategies of the form λi , (Fix )x∈Supp(λi ) , where λi ∈ ∆(X) and Fix is a cdf over [0, 1] for every x ∈ Supp(λi ). We refer to λi as firm i’s framing strategy and to Fix as firm i’s pricing strategy at x. Let µx ∈ ∆(X) denote a degenerate probability distribution that assigns probability one to node x. The marginal pricing ¢ ¡ strategy induced by a mixed strategy λi , (Fix )x∈Supp(λi ) is Fi =

X

λi (x)Fix

x∈Supp(λ)

Given a cdf F on [0, 1], let F − denote its left limit. Given a realization (pi , xi )i=1,2 of the firms’ strategies, the consumer chooses a firm according to the following rule. He is randomly assigned to a firm - with probability 1 for each firm. Suppose that he is assigned to firm i. If there is a direct link from xi 2 to xj - an event that occurs with probability π(xi , xj ) - the consumer makes a price comparison and chooses firm j if pj < pi . Otherwise, the consumer chooses the initially assigned firm i. The consumer’s initial assignment to a firm can be interpreted as the first alternative considered in a sequential decision process or as a default option arising from previous decisions. The consumer’s choice procedure is biased in favor of the initial firm i: the consumer selects it with probability one when pj ≥ pi and with probability 1 − π(xi , xj ) when pj < pi . When the graph is order-independent, the sequential aspect of the choice procedure is inessential. In this case, the model is consistent with an additional interpretation in which the consumer is confronted with both alternatives simultaneously, chooses the cheaper one if the formats are linked, and chooses randomly otherwise. To illustrate the firms’ payoff function, consider the graph in Figure 2. Thus, π (x, y) = q and there is no link from y to x. Suppose that firm 1 adopts the format x while firm 2 adopts the format y. If p1 < p2 , firm 1 earns a payoff of 12 p1 while firm 2 earns 12 p2 . If p1 > p2 , firm 1 earns p1 · ( 12 − 12 q) while firm 2 earns p2 · ( 12 + 12 q).

8

x

q

y

Figure 2 ¡ ¢ When firm i plays the mixed strategy λi , (Fix )x∈Supp(λi ) , we can write firm j’s expected payoff from the pure strategy (p, x) as follows: X p λi (y) · [(1 − Fiy (p)) · π(y, x) − Fiy− (p) · π(x, y)]} · {1 + 2 y∈X Is consumer choice rational? Fully rational consumers are represented by a complete graph - i.e. π(x, y) = 1 for all x, y ∈ X. Rational consumers make a price comparison independently of the firms’ framing decisions, and in this case the model is reduced to standard Bertrand competition. For a typically incomplete graph, the consumer’s choice behavior is inconsistent with maximizing a random utility function over price-format pairs. To see why, consider the following deterministic, order-independent graph: X = {a, b, c}, π(a, b) = π(b, c) = 1 and π(a, c) = 0. Suppose that p < p0 < p00 . When faced with the strategy profile ((p, a), (p0 , b)), the consumer chooses (p, a) with probability one. Similarly, when faced with the strategy profile ((p0 , b), (p00 , c)), the consumer chooses (p0 , b) with probability one. However, when faced with the strategy profile ((p, a), (p00 , c)), the consumer chooses each alternative with probability 12 . No random utility function over [0, 1] × X can rationalize such choice behavior. The reason is that the graph represents a binary relation which is intransitive, and this translates into the intransitivity of the implied revealed preference relation over price-format pairs. Hide and seek Our analysis will make use of an auxiliary two-player, zero-sum game, which is a generalization of familiar games such as Matching Pennies. The players are referred to as hider and seeker, denoted h and s. The players share the same action space X. Given the action profile (xh , xs ), the hider’s payoff is −π(xh , xs ) and the seeker’s payoff 9

is π(xh , xs ). We will refer to this game as the hide-and-seek game associated with a graph. Given a mixed-strategy profile (λh , λs ) in this game, the probability that the seeker finds the hider is v (λh , λs ) =

XX

λh (x) λs (y) π (x, y)

x∈X y∈X

To see the relevance of this auxiliary game to our model, suppose that firm 1 plays a mixed strategy with framing strategy λ and an atomless marginal pricing strategy F over the support [pL , pU ]. When firm 2 considers charging the price pU , it should select a format that minimizes the probability of a price comparison. Hence, it behaves as a hider in the hide-and-seek game, where the seeker’s strategy is λ. Similarly, when firm 2 considers charging the price pL , it should select a format that maximizes the probability of a price comparison. Hence, it behaves as a seeker in the hide-and-seek game, where the hider’s strategy is given by λ. When a firm considers charging an intermediate price, it chooses its framing strategy partly as a hider and partly as a seeker. The value of the hide-and-seek game is v∗ = max min v (λh , λs ) λs

λh

The max-min payoff of a firm in our model is thus 12 (1 − v ∗ ). The reason is that the worst-case scenario for a firm is that its opponent plays p = 0 and adopts the seeker’s max-min framing strategy, to which a best-reply is to play p = 1 and minimize the probability of a price comparison. Preliminary analysis of Nash equilibria We will conduct a detailed analysis of Nash equilibria in the following sections. In this section, we present two basic results. The first characterizes the support of the marginal pricing strategies when both firms make positive profits. The second provides a simple necessary and sufficient condition for the equilibrium outcome to be competitive (that is, both firms charge zero prices).

Proposition 1 In any Nash equilibrium in which firms make positive profits, there exists a price pl ∈ (0, 1) such that, for i = 1, 2: (i) the support of Fi is [pl , 1]; (ii) Fi is strictly increasing on [pl , 1]. 10

Proposition 2 Let Fi be a Nash equilibrium marginal pricing strategy for firm i, i = 1, 2. Then, F1 (0) = F2 (0) = 1 if and only if there exists a format x∗ ∈ X such that π(x, x∗ ) = 1 for every x ∈ X. Note that a corollary of Proposition 1 is that if firm i earns the max-min payoff − v ∗ ) in Nash equilibrium, then it must be the case that firm j’s framing strategy conditional on p < 1 is a max-min strategy for the seeker in the associated hide-andseek game. The proofs of these results rely on price undercutting arguments that are somewhat more subtle than familiar ones. For instance, suppose that firm 1’s marginal pricing strategy has a mass point at some price p∗ which belongs to the support of firm 2’s marginal pricing strategy. In conventional models of price competition, there is a clear incentive for firm 2 to undercut its price slightly below p∗ . In our model, however, price undercutting may have to be accompanied by a change in the framing strategy in order to be effective. Adopting a new framing strategy may be undesirable for firm 2 because it could change the probability of a price comparison when the realization of firm 1’s pricing strategy is p 6= p∗ . For the rest of the paper, we assume that the necessary and sufficient condition for a competitive equilibrium outcome is violated. 1 (1 2

Condition 1 For every x ∈ X there exists y 6= x such that π(y, x) < 1. This condition ensures that the firms’ max-min payoff is strictly positive - or, equivalently, that the value of the associated hide-and-seek game is strictly below one. Once competitive equilibrium outcomes have been eliminated, any Nash equilibrium must be mixed. To see why, assume that each firm i plays a pure strategy (pi , xi ). If 0 < pi ≤ pj , then firm j can deviate to the strategy (pi − ε, xi ), where ε > 0 is arbitrarily small, and raise its payoff. If pi = 0, firm i earns zero profits, contradicting the observation that the firms’ max-min payoffs are strictly positive. Thus, from now on, we will take it for granted that Nash equilibrium is strictly mixed.

3

Nash Equilibrium under Order Independence

In this section, we analyze mixed strategy equilibria in order-independent graphs. We present the notion of weighted regularity, some general characterization results, 11

and a complete characterization of symmetric equilibria in the class of so-called “bisymmetric” graphs. We use this characterization to highlight the non-trivial effects that greater consumer rationality has on equilibrium prices in our model. Finally, we discuss the novel account that our model provides for the phenomenon of product differentiation.

3.1

Weighted Regularity

P When an order-independent graph is regular - i.e. when y∈X π (x, y) = v¯ for all nodes x ∈ X - all formats are equally comparable in the sense that each format has an identical expected number of links. However, this notion of equal comparability ignores the frequency with which different formats are adopted. If, for example, x is an isolated node and both firms choose this format, the consumer will make a price comparison with probability one. Hence, a proper notion of equal comparability should take into account the frequency of adoption of different formats. Definition 1 An order-independent graph (X, π) is weighted-regular if there exist β ∈ P ∆(X) and v¯ ∈ [0, 1] such that y∈X β (y) π (x, y) = v¯ for any x ∈ X. We say that β verifies weighted regularity. Regularity thus corresponds to a special case in which the uniform distribution over X verifies weighted regularity. Note that the set of distributions that verify weighted regularity is convex. The following are examples of weighted-regular, order-independent graphs. Example 3.1: Equivalence relations. Consider a deterministic graph that in which π(x, y) = 1 if and only if x ∼ y, where ∼ is an equivalence relation. Any distribution that assigns equal probability to each equivalence class verifies weighted regularity. Example 3.2: A cycle with random links. Let X = {1, 2, ..., n}, where n is even. Assume that for every distinct x, y ∈ X, π(x, y) = 12 if |y − x| = 1 or |y − x| = n − 1, and π(x, y) = 0 otherwise. A uniform distribution over all odd-numbered nodes verifies weighted regularity. Example 3.3: Linear similarity. Consider the following deterministic graph. Let X = {1, 2, ..., 3L}, where L ≥ 2 is an integer. For every distinct x, y ∈ X, π(x, y) = 1 if and only if |x − y| = 1. A uniform distribution over the subset {3k − 1}k=1,...,L verifies weighted regularity. 12

In addition, note that the graph given by Figure 1 is weighted regular if and only if q = 0. The framing strategy that verifies weighted regularity in this case assigns probability 12 to each of the two core nodes. Lemma 1 The distribution λ ∈ ∆(X) verifies weighted regularity in a graph (X, π) if and only if (λ, λ) is a Nash equilibrium in the associated hide-and-seek game. Proof. Suppose that λ verifies weighted regularity. If one of the players in the associated hide-and-seek game plays λ, every strategy for the opponent - including λ itself - is a best-reply. Now suppose that (λ, λ) is a Nash equilibrium in the associated hide-and-seek game. Denote v(λ, λ) = v∗ . If some format attains a higher probability of a price comparison than v∗ , then λ cannot be a best-reply for the seeker. Similarly, if some format attains a lower probability of a price comparison than v∗ , then λ cannot be a best-reply for the hider. Therefore, it must be the case that every format generates the same probability of a price comparison - namely v∗ - against λ. Thus, a graph is weighted-regular if and only if the associated hide-and-seek game has a symmetric Nash equilibrium.

3.2

Price-Format Independence and Equilibrium Payoffs

¡ ¢ A mixed strategy λ, (F x )x∈Supp(λ) exhibits price-format independence if F x = F y for any x, y ∈ Supp(λ). The next proposition shows that if a graph is weighted-regular, there exists a symmetric Nash equilibrium that exhibits price-format independence. Conversely, if the strategies are price-format independent in some Nash equilibrium, then each firm plays a framing strategy that verifies weighted regularity and earns max-min payoffs. In addition, the firms’ pricing strategies must be identical. Define the cdf 1 − v∗ 1 − p (1) · G∗ (p) = 1 − 2v ∗ p with support [

1 − v∗ , 1]. 1 + v∗

Proposition 3 Consider a graph (X, π). (i) Suppose that λ1 and λ2 verify weighted regularity. Then, there exists a Nash equilibrium in which firm i, i = 1, 2, plays the framing strategy λi and the pricing strategy Fix (p) = G∗ (p) for all x ∈ X, and earns max-min payoffs. 13

¡ ¢ (ii) Let λi , (Fix )x∈Supp(λi ) i=1,2 be a Nash equilibrium in which both firms’ strategies exhibit price-format independence. Then, λ1 and λ2 verify weighted regularity, firms earn max-min payoffs, and their marginal pricing strategy is given by 1. Proof. (i) Suppose that firm i plays the framing strategy λi . By the definition of weighted regularity, every format that the rival firm j may adopt attains the same probability of a price comparison v∗ against λi . We can thus assume that the probability of a price comparison is exogenously fixed at v ∗ . The pricing strategy F given by (1) has the property that for every p in the support of F , 1 − v∗ p = · [1 + v ∗ (1 − F (p)) − v∗ F (p)] 2 2 which is necessary and sufficient for F to be a best-replying pricing strategy to itself, given that the probability of a price comparison is v∗ . (ii) By assumption, Fix = Fi for any x ∈ Supp(λi ), i = 1, 2. Therefore, x ∈ arg min v(·, λi ) for every x ∈ Supp(λj ) - otherwise, it would be profitable to deviate to the pure strategy (1, y) for some y ∈ arg min v(·, λi ). Similarly, x ∈ arg max v(·, λi ) for every x ∈ Supp(λj ) - otherwise, it would be profitable to deviate to the pure strategy (pl , y) for some y ∈ arg max v(·, λi ). It follows that (λ1 , λ2 ) and (λ2 , λ1 ) are Nash equilibria of the associated hide-and-seek game. Hence, as λ1 and λ2 maxminimize as well as min-maximize v, (λ1 , λ1 ) and (λ2 , λ2 ) are also Nash equilibria of the associated hide-and-seek game. Therefore, both λ1 and λ2 verify weighted regularity. Relatively standard arguments (see Proposition 1 in Spiegler (2006)) establish that the equilibrium pricing strategy for each firm must be given by (1) if the probability of a price comparison is exogenously fixed at v∗ . The intuition behind this result is simple. When firms play framing strategies that verify weighted regularity, their opponents are indifferent among all formats and can treat the probability of a price comparison v∗ as fixed and exogenous. Therefore, we can construct an equilibrium in which firms play framing strategies that verify weighted regularity, and an independent pricing strategy. For the converse, note that the framing strategy associated with the highest price in the equilibrium distribution minimizes the probability of a price comparison against the equilibrium framing strategy. Similarly, the framing strategy associated with the lowest price in the equilibrium distribution maximizes the probability of a price comparison against the equilibrium framing strategy. If these two framing strategies coincide, then all formats must induce the same probability of a price comparison against the opponent’s framing strategy. 14

To demonstrate this result, let us revisit some of the examples presented in the previous sub-section. In Example 3.2, suppose that firm 1 (2) plays a framing strategy which is a uniform distribution over all odd-numbered (even-numbered) nodes. Both distributions verify weighted regularity. Suppose further that both firms play independently the pricing strategy given by (1), where v∗ = n2 . This strategy profile constitutes a Nash equilibrium. In Example 3.3, suppose that both firms play a framing strategy which mixes uniformly over the subset of nodes {3k − 1}k=1,...,L . This distribution verifies weighted regularity. Suppose further that both firms play independently the pricing strategy given by (1), where v∗ = L1 . This strategy profile constitutes a symmetric Nash equilibrium, in which the consumer makes a price comparison if and only if the firms adopt the same format. In this equilibrium, the formats played with positive probability are “local monopolies”: when the consumer faces two different formats, he remains loyal to the one adopted by the firm he is initially assigned to. Price comparisons take place only when both firms use the same format. Not all Nash equilibria in weighted-regular graphs necessarily exhibit price-format independence. This is trivially the case in graphs that contain redundant nodes (i.e., there exist distinct formats x, x0 such that π(x, y) = π(x0 , y) for every y ∈ X). In this case, we can construct an equilibrium in which the framing strategy verifies weighted regularity, yet the format x is associated with low prices while the format x0 is associated with high prices. As we will see in Section 3.3, price-format correlation is possible under weighted-regular graphs even when there are no redundant nodes. Two questions are still open. Do max-min equilibrium payoffs imply that the graph is weighted-regular? Does weighted regularity imply that equilibrium payoffs cannot exceed the max-min level? We are only able to address these questions under some restrictions on equilibrium strategies. ¢ ¡ Proposition 4 Consider a Nash equilibrium λi , (Fix )x∈Supp(λi ) i=1,2 . If firm 1 earns max-min payoffs and firm 2 plays a framing strategy with full support, then (X, π) is weighted-regular. Proof. The proof is based on the following version of Farkas’ lemma. Let Ω be an l × m matrix and b an l-dimensional vector. Then, exactly one of the following two statements is true: (i) there exists β ∈ Rm such that Ωβ = b and β ≥ 0; (ii) there exists δ ∈ Rl such that ΩT δ ≥ 0 and bT δ < 0.

15

Suppose that (X, π) is not weighted-regular. Let us first show that for every µ ∈ ∆(X) such that µ (x) > 0 for all x ∈ X, there exists µ ˜ ∈ ∆(X) such that, for all y ∈ X, X

µ (x) π (x, y) <

x∈X

X

µ ˜ (x) π (x, y)

x∈X

Order the nodes so that X = {1, .., n}. Any β ∈ ∆(X) is thus represented by a row vector (β 1 , ..., β n ). Let Π be a n × n matrix whose ijth entry is π (i, j). Note that Π = ΠT . Since (X, π) is not weighted-regular, there exist no β ∈ Rn and c > 0 such that Πβ T = (c, c, ..., c)T . By Farkas’ Lemma, there exists a column vector δ ∈ Rn such that Πδ ≥ 0 and (c, c, ..., c)δ < 0. Since π(i, i) = 1 for every i ∈ {1, ..., n} and π(i, j) ≥ 0 for all i, j ∈ {1, ..., n}, we can modify δ into a column vector ˜δ such that ˜δ i > δ i for every i, Π˜δ > 0 and P ˜δi = 0. Let µ ∈ ∆(X) and µ(i) > 0 for every i i ∈ {1, ..., n}. By the construction of ˜δ, µ ˜ = µ + α˜δ is also a probability distribution over X, for a sufficiently small α > 0. Then Π˜ µT = ΠµT + αΠ˜δ > ΠµT

In particular, every component of the vector Π˜ µT is strictly larger than the corresponding component of ΠµT . By hypothesis, λ2 (x) > 0 for all x ∈ X. We have shown that there exists another ˜ such that every format y ∈ X induces a strictly higher probability framing strategy λ of a price comparison than λ2 . This contradicts that λ2 is a max-min strategy. The proof of this result relies entirely on the associated hide-and-seek game. It shows that if, in the hide-and-seek game, there exists a max-min strategy with full support for the seeker, there must exist a symmetric Nash equilibrium.

3.3

Cutoff Equilibria

In this sub-section, we study equilibria that exhibit a simple kind of price-format correlation. A symmetric Nash equilibrium in which firms play the strategy (λ, (F x )x∈Supp(λ) ) is a cutoff equilibrium if there exist prices pl ≤ pm ≤ 1 such that for all x ∈ Supp(λ), the support of F x is either [pl , pm ] or [pm , 1]. Thus, in a cutoff equilibrium formats are unambiguously associated either with high prices or with low prices. Let λU be the framing strategy conditional on the nodes in which the pricing strategy has support [pm , 1]. Similarly, let λL be the framing strategy conditional on the nodes in which the 16

pricing strategy has support [pl , pm ]. Using standard arguments, it can be easily shown that in a cutoff equilibrium F x is continuous for any x ∈ Supp(λ) due to symmetry. Define α = 1 − F (pm ). Lemma 2 If (λ, (F x )x∈Supp(λ) ) is a cutoff equilibrium strategy with pl < pm < 1, then (λU , λL ) is a Nash equilibrium in the associated hide-and-seek game. Proof. Let (λ, (F x )x∈Supp(λ) ) be a cutoff equilibrium strategy. Note that a firm charging pm is indifferent between λU and λL . Moreover, λU minimizes v(λU , λ) and λL maximizes v(λL , λ). Since λ = αλU + (1 − α)λL , the payoff from the strategy (pm , λU ) can be written as pm (1 + v(λU , λ) − 2 (1 − α) v(λU , λL )) 2 Since λU minimizes v(·, λ), it must be the case that λU minimizes v(·, λL ). The payoff from the strategy (pm , λL ) can be written as pm (1 + 2αv(λL , λU ) − v(λL , λ)) 2 Since λL maximizes v(·, λ), it must be the case that λL maximizes v(·, λU ). Hence, (λL , λU ) is a Nash equilibrium in the hide-and-seek game. Corollary 1 If (λ, (F x )x∈Supp(λ) ) is a cutoff equilibrium strategy with pl < pm < 1, then: (2) αv(λU , λU ) + (1 − α)v(λL , λL ) = v ∗ and v(λU , λU ) ≤ v ∗ ≤ v(λL , λL )

(3)

Proof. In a cutoff equilibrium, every node in the supports of λU and λL is optimal at the cutoff price pm . This implies αv(λU , λU ) − (1 − α)v(λU , λL ) = αv(λL , λU ) − (1 − α)v(λL , λL ) By Lemma 2, equation (2) follows immediately. Since λU minimizes v(·, λ) and λL maximizes v(·, λ), we have v(λU , λ) = αv(λU , λU ) + (1 − α)v(λU , λL ) ≤ αv(λL , λU ) + (1 − α)v(λL , λL ) = v(λL , λ) 17

Combining this with (2), we obtain (3). Proposition 5 Suppose that a graph (X, π) admits a cutoff equilibrium in which both firms play the strategy (λ, (F x )x∈Supp(λ) ). (i) If firms earn max-min payoffs, then λ verifies weighted regularity. (ii) If the graph is weighted-regular, then firms earn max-min payoffs. Proof. (i) Assume that firms earn max-min payoffs. If pm = pl or pm = 1, then arg minx∈X v(x, λ) = arg maxx∈X v(x, λ). Hence λ verifies weighted regularity. Thus, suppose that pl < pm < 1. Since λ max-minimizes v, λ is a max-min strategy for the seeker in the associated hide-and-seek game. By Lemma 2, (λU , λL ) is a Nash equilibrium in the hide-and-seek game. Therefore, v(λU , λL ) = v(λU , λ) = v∗ . Since v(λU , λ) = αv(λU , λU ) + (1 − α)v(λU , λL ) we have v(λU , λU ) = v ∗ . By equation (2), v(λL , λL ) = v∗ , and hence v(λL , λ) = v ∗ . Since λU minimizes v(·, λ) and λL maximizes v(·, λ), the claim follows. (ii) Suppose that the graph (X, π) is weighted-regular, and let β ∈ ∆(X) be a framing strategy that verifies this property. Therefore, v(x, β) = v ∗ for every x ∈ X. Now suppose that (λ, (F x )x∈Supp(λ) ) is a cutoff equilibrium. Since λU and λL are optimal at pm : 2αv(λU , λU ) − v(λU , λ) ≥ 2αv ∗ − v∗

2αv(λL , λU ) − v(λL , λ) ≥ 2αv ∗ − v∗ .

By Lemma 2, these inequalities yield v(λU , λU ) ≥ v ∗ ≥ v(λL , λL ) which together with (3) yield v(λU , λU ) = v(λL , λL ) = v∗ By definition, v(λU , λ) = αv(λU , λU ) + (1 − α)v(λU , λL ) v(λL , λ) = αv(λL , λU ) + (1 − α)v(λL , λL ) 18

Therefore, v(λU , λ) = v(λL , λ). Hence, λ verifies weighted regularity and firms earn max-min payoffs. The intuition for this result is as follows. According to Lemma 2, the formats adopted in the low (high) price range of a cutoff equilibrium are “seeking formats” (“hiding formats”) that aim to maximize (minimize) the probability of a price comparison. When weighted regularity is violated, there is a real distinction between seeking and hiding formats. When both firms realize a price in the high range, the probability that the consumer chooses correctly is relatively low, because the firms’ framing strategy conditional on p > pm evades a price comparison. In particular, when a firm charges the monopolistic price p = 1 it is compared to the rival firm with a probability below v ∗ , hence its payoff exceeds the max-min level. Thus, the distinction between “seeking” and “hiding” formats gives firms a market power they lack when the graph is weighted-regular (where “seeking” and “hiding” formats induce the same probability of a price comparison). The illustrative example in the Introduction demonstrates this effect.

Figure 3 For a non-trivial example of a weighted-regular graph that gives rise to a cutoff equilibrium, consider the deterministic, nine-node graph given by Figure 3. A uniform distribution over the six bold nodes verifies weighted regularity (v∗ = 13 ). One can construct an equilibrium in which this is indeed the framing strategy, and yet framing and pricing decisions are correlated. Specifically, the three peripheral nodes are played with probability 13 each conditional on p ∈ [ 23 , 1], while their internal neighbors are played with probability 13 each conditional on p ∈ [ 12 , 23 ). The marginal pricing strategy is given by expression (1). 19

3.4

Bi-Symmetric Graphs

In this sub-section, we provide a complete characterization of symmetric Nash equilibrium in a special class of graphs. An order-independent graph (X, π) is bi-symmetric if X can be partitioned into two sets, Y and Z, such that for every distinct x, y ∈ X:    qY if x, y ∈ Y , x 6= y π(x, y) = qZ if x, y ∈ Z, x 6= y   q if x ∈ Y and y ∈ Z

where max{qY , qZ , q} < 1. One natural interpretation is that Y and Z represent two broad ways of spuriously categorizing products. Under this interpretation, it makes sense to assume that two particular brands are more comparable when they are similarly categorized - i.e., q ≤ min{qY , qZ }. In contrast, when qY ≤ q ≤ qZ , it is more natural to interpret Y and Z as two broad price formats, where Y represents a more complex format than Z. Define 1 + qI · (nI − 1) qI∗ = nI where I = Y, Z and nI = |I|. Without loss of generality, assume qZ∗ ≥ qY∗ . One can verify (see the proof of Proposition 6 in the Appendix) that a bi-symmetric graph is weighted-regular if and only if (qY∗ − q)(qZ∗ − q) ≥ 0 When qY∗ = qZ∗ = q, there is a continuum of framing strategies that verify weighted regularity. Otherwise, the unique framing strategy that verifies weighted regularity qY∗ − q to the set Z, and mixes uniformly within Y assigns probability ∗ (qY − q) + (qZ∗ − q) and within Z. The value of the hide-and-seek game under weighted regularity is

v∗ =

        

q

when qY∗ = qZ∗ = q

qY∗ qZ∗ − q 2 (qY∗ − q) + (qZ∗ − q)

(4) otherwise

By Proposition 3, if weighted regularity holds, any distribution that verifies weighted 20

regularity is an equilibrium framing strategy and G∗ (p) given in (1) is a price-format independent equilibrium pricing strategy. When the condition for weighted regularity is not satisfied - i.e., when q is strictly between qY∗ and qZ∗ - the value of the game is v∗ = q, since there is a Nash equilibrium in the hide-and-seek game in which the seeker plays U(Z), the uniform distribution over Z, while the hider plays U(Y ), the uniform distribution over Y . It can be verified that there exists a cutoff equilibrium in which: λU ≡ U(Y )

(5)

λL ≡ U(Z) q − qY∗ m F (p ) = ∗ qZ − qY∗ Recall that α = 1 − F (pm ). Applying equation (2), we obtain the following expression for the firms’ equilibrium payoff: qZ∗ − q 1 q − qY∗ ·[ ∗ · (1 − q) + · (1 − qY∗ )] ∗ ∗ ∗ 2 qZ − qY qZ − qY which strictly exceeds the max-min payoff 12 (1 − q). The pricing strategy can be easily derived from (5). We omit it for brevity. The following proposition states that the above equilibria characterize the set of symmetric equilibria. Proposition 6 Let (X, π) be a bi-symmetric graph. In any symmetric Nash equilibrium: (i) If (qY∗ −q)(qZ∗ −q) ≥ 0, firms play a framing strategy that verifies weighted regularity. If (qY∗ − q)(qZ∗ − q) > 0, the pricing strategy at each x ∈ X is given by (1), where v∗ is given by (4). (ii) If (qY∗ − q)(qZ∗ − q) < 0, firms play the cutoff equilibrium given by (5). This result provides another illustration for the non-trivial relation between consumer rationality and equilibrium profits. Imagine a regulator who wishes to impose a disclosure policy that will enhance the comparability of different price formats. Suppose that qY∗ < q < qZ∗ . If the regulator’s intervention increases the values of q and qY∗ , the intervention will cause expected price to go down and thus benefit the consumer. If, however, the intervention causes an increase in the value of qZ∗ (without changing 21

q and qY∗ ), the intervention will hurt the consumer. As in the example presented in the Introduction, a higher qZ∗ gives relatively expensive firms a stronger incentive to adopt “hiding” formats in Y . Since q and qY∗ are unchanged, the probability that an expensive firm faces a price comparison with its opponent goes down. As a result of this greater effective market power, firms earn higher profits in equilibrium.

3.5

Symmetric Graphs

Nash equilibrium is not necessarily unique and not necessarily symmetric in our model. Recall that in Example 3.2, there exist asymmetric mixed-strategy equilibria, in which firms randomize over disjoint sets of formats. However, in these equilibria, the firms’ pricing strategies and profits are the same as in the symmetric equilibrium. Whether this is a general property of equilibria in our model is an open question. For a special class of graphs, we are able to establish the uniqueness of Nash equilibria. We say that a graph (X, π) is symmetric if π (x, y) = q for all distinct x and y. Proposition 7 Suppose that (X, π) is symmetric with q < 1. The Nash equilibrium is unique. Both firms play the framing strategy U(X). Moreover, Fix is given by (1) for every x ∈ X, i = 1, 2, where 1 + q (n − 1) v∗ = (6) n Thus, framing asymmetries across firms and price levels are impossible in equilibrium. Note that symmetric graphs are a special case of bi-symmetric graphs in which qY = qZ = q. A graph representing an equivalence relation with K equivalence classes is essentially identical to a symmetric graph with n = K and q = 0 where each node represents an equivalence class. Proposition 7 thus implies that in every Nash equilibrium, firms play a framing strategy that randomizes uniformly over the K equivalence classes. Moreover, for every equivalence class k the firms’ pricing strategy conditional on k is given by (1), where v ∗ is given by (6) with n replaced by K.

4

Relaxing Order Independence

In this section we explore some properties of Nash equilibria when a graph violates order independence. We begin by extending the notion of weighted regularity. 22

Definition 2 A graph (X, π) is weighted-regular if there exist β ∈ ∆(X) and v¯ ∈ [0, 1] P P such that y∈X β (y) π (x, y) = y∈X β (y) π (y, x) = v¯ for all x ∈ X. We say that β verifies weighted regularity. The equivalence between weighted regularity and the existence of symmetric equilibrium in the associated hide-and-seek game, established for order-independent graphs, needs to be qualified when order independence is relaxed. Lemma 3 (i) If λ verifies weighted regularity, then (λ, λ) is a Nash equilibrium in the hide-and-seek game; (ii) If (λ, λ) is a Nash equilibrium in the hide-and-seek game and λ(x) > 0 for every x ∈ X, then λ verifies weighted regularity. Proof. The proof of part (i) is identical to the order-independent case. Let us turn to part (ii). Suppose that (λ, λ) is a Nash equilibrium in the hide-and-seek game. Since λ is a best-reply for the hider against λ, v(µx , λ) ≥ v(λ, λ) for every x ∈ X. By the full-support assumption, if there is a frame x ∈ X for which v(µx , λ) > v(λ, λ), then P x x∈X λ(x)v(µ , λ) > v(λ, λ). The L.H.S. of this inequality is by definition v(λ, λ), a contradiction. Similarly, since λ is a best-reply for the seeker against λ, v(λ, µx ) ≤ v(λ, λ) for every x ∈ X. By the full-support assumption, if there is a frame x ∈ X for P which v(λ, µx ) < v(λ, λ), then x∈X λ(x)v(λ, µx ) < v(λ, λ), a contradiction. It follows that for every x ∈ X, v(µx , λ) = v(λ, µx ) = v(λ, λ). To see how the full support assumption is necessary for the second part of this lemma, consider the deterministic graph given by Figure 4. The hide-and-seek game induced by this graph has a symmetric Nash equilibrium in which both the hider and the seeker play y and z with probability 12 each. However, the graph is not weightedregular.

y

z

x

Figure 4 23

The full-support qualification carries over to the next result, which is a variation on Proposition 3. Proposition 8 (i) Suppose that λ1 and λ2 verify weighted regularity. Then, there exists a Nash equilibrium in which each firm i = 1, 2 plays the framing strategy λi and the pricing strategy Fix (p) = G∗ (p) for all x ∈ X, and earns max-min payoffs. ¢ ¡ (ii) Let λi , (Fix )x∈Supp(λi ) i=1,2 be a Nash equilibrium in which the pricing strategies exhibit price-format independence and the framing strategies have full support for both firms. Then, λ1 and λ2 verify weighted regularity, firms earn max-min payoffs, and their marginal pricing strategy is given by (1). Proof. Analogous to the proof of Proposition 3. One can extend the notion of bi-symmetric graphs by allowing asymmetric connectivity between the sets Y and Z - that is, π(y, z) = qY Z and π(z, y) = qZY for every y ∈ Y , z ∈ Z, where qY Z 6= qZY (while maintaining the assumption that connectivity is symmetric and constant within each of the two sets). The reader can easily verify that such graphs are never weighted regular. The following example demonstrates that they admit a type of price-format correlation different from the one captured by cutoff equilibria. In particular, the supports of the pricing strategies are nested. Consider the graph in Figure 2: X = {x, y}, π (x, y) = q and π (y, x) = 0. There is a symmetric Nash equilibrium in which the firms play a framing strategy that satisfies , and a pricing strategy given by: λ (x) = 1−q 2−q 1 (3p + q − 1) 2p + pq 1 F x (p) = (3p + pq − 1) 2p + pq F y (p) =

1 , 1], and over the interval [ 3+q

¢ 1 ¡ 3p + q − pq 2 − 1 2p x F (p) = 0 F y (p) =

1−q 1 over the interval [ 3−q 2 , 3+q ]. Note that firms earn max-min payoffs in this equilibrium.

An incumbent-entrant model

24

Equilibrium analysis under order dependence is greatly simplified if the assumption that the consumer’s initial firm assignment is random is dropped. Suppose that the consumer is initially assigned to firm 1, referred to as the Incumbent. Firm 2 is referred to as the Entrant. In this case, firm 1’s max-min payoff is 1−v∗ , while firm 2’s max-min payoff is zero. ¢ ¡ Proposition 9 Any Nash equilibrium λi , (Fix )x∈Supp(λi ) i=1,2 of the Incumbent-Entrant model has the following properties: (i) (λ1 , λ2 ) constitutes a Nash equilibrium in the associated hide-and-seek game in which firm 1 (2) is the hider (seeker). (ii) Firm 1’s equilibrium payoff is 1 − v ∗ while firm 2’s equilibrium payoff is v∗ (1 − v∗ ). (iii) The firms’ marginal pricing strategies over [1 − v ∗ , 1) are given by: 1 − v∗ p 1 1 − v∗ ] F2 (p) = ∗ · [1 − v p F1 (p) = 1 −

and F1 has an atom of size 1 − v∗ at p = 1. The simplicity of the equilibrium characterization in this case results from the firms’ unambiguous incentives when choosing their framing strategies. The Incumbent has an unequivocal incentive to avoid a price comparison (because then it is chosen with probability one), while the Entrant has an unequivocal incentive to enforce a price comparison (because otherwise it is chosen with probability zero).

5

Remarks on Product Differentiation

The firms’ mixing over formats in Nash equilibrium of our model can be viewed as a type of product differentiation. Since in our model the firms’ product is inherently homogenous, differentiation is a pure reflection of the firms’ attempt to avoid price comparisons. By comparison, in conventional models product differentiation is viewed as the market’s response to consumers’ differentiated tastes. It may be useful to think of our model in terms of spatial competition. Think of firms as stores and of nodes as possible physical locations of stores. A link from one location x to another location y indicates that it is costless to travel from x to y. The absence of a link from x to y means that it is impossible to travel in this direction. 25

According to this interpretation, the consumer follows a myopic search process in which he first goes randomly to one of the two stores (independently of their locations). Then, he travels to the second store if and only if the trip is costless. Finally, the consumer chooses the cheaper firm that his search process has elicited (with a tie-breaking rule that favors the initial firm.)3 Although this re-interpretation is reminiscent of the literature on spatial competition, there is a crucial difference. In conventional models of spatial competition, consumers are attached to specific locations and choose between stores according to their price and the cost of travelling to their location. In particular, a consumer who is attached to a location x does not care at all about the cost of transportation between two stores if neither of them is located at x. In contrast, in our model, consumer choice is always sensitive to the probability of a link between the firms’ locations. Recall that in our model consumer choice is typically impossible to rationalize with a random utility function over pairs (p, x). In contrast, conventional models of spatial competition (and product differentiation in general) are based on the assumption that consumer choice is consistent with a random utility function over price-location pairs. The different consumer behavior in these two classes of models generates differences in equilibrium outcomes. First, recall our observation in Section 2 that our model does not admit pure-strategy Nash equilibria that support non-zero prices. Second, consider the following spatial competition model that fits the graph of Figure 1. The consumer is attached to each core node with probability α and to each peripheral node with probability (1 − 2α)/4. He chooses the cheapest firm among those whose location is linked to his own location (with a symmetric tie-breaking rule). It can be shown that in symmetric equilibrium of this model, firms assign zero probability to the peripheral nodes for every value of α and q. These two approaches to product differentiation can be conveniently synthesized. Suppose that a consumer type θ is characterized by two primitives: a graph π θ and a willingness-to-pay function uθ : X → {0, 1}. The function uθ essentially describes the set of product formats (or brands) that type θ likes, whereas the graph π θ determines the type’s ability to make price comparisons. Exploring this synthesized model is left for future work. 3

This search process is unconventional in two ways. First, whether or not it is costly for the consumer to examine the second firm depends on both firms’ strategic decisions (namely their choice of formats). Second, the consumer in our model does not design his sequential search optimally according to “rational expectations” of the firms’ behavior.

26

References [1] Anderson, S., A. de Palma and J.F. Thisse (1992): Discrete Choice Theory of Product Differentiation, MIT Press. [2] Ellison, G. (2006): “Bounded Rationality in Industrial Organization,” in Richard Blundell, Whitney Newey and Torsten Persson (eds.), Advances in Economics and Econometrics: Theory and Applications, Ninth World Congress, Cambridge University Press. [3] Eliaz, K. and R. Spiegler (2007): “Consideration Sets and Competitive Marketing,” mimeo. [4] Eliaz, K. and R. Spiegler (2008): “Consumer Optimism and Price Discrimination,” Theoretical Economics 3, 459-497. [5] Gabaix, X. and D. Laibson (2006): “Shrouded Attributes, Consumer Myopia, and Information Suppression in Competitive Markets,” Quarterly Journal of Economics 121, 505-540. [6] Nedungadi, P. (1990): “Recall and Consumer Consideration Sets: Influencing Choice without Altering Brand Evaluations,” Journal of Consumer Research, 17, 263-276. [7] Piccione, M. and A. Rubinstein (2003): “Modeling the Economic Interaction of Agents with Diverse Abilities to Recognize Equilibrium Patterns,” Journal of European Economic Association 1, 212-223. [8] Rubinstein, A. (1993): “On Price Recognition and Computational Complexity in a Monopolistic Model,” Journal of Political Economy 101, 473-484. [9] Rubinstein, A. and Y. Salant (2008): “(A,f): Choices with Frames,” Review of Economic Studies.75, 1287-1296. [10] Spiegler, R. (2005): “The Market for Quacks,” Review of Economic Studies 73, 1113-1131. [11] Spiegler R. (2006): “Competition over Agents with Boundedly Rational Expectations,” Theoretical Economics 1, 207-231. [12] Varian, H. (1980): “A Model of Sales,” American Economic Review 70, 651-59.

27

6 6.1

Appendix: Proofs Proposition 1

Consider a Nash equilibrium in which firms earn strictly positive payoffs. For each firm i = 1, 2, let pli denote the infimum of the support of Fi . Clearly, pl1 = pl2 . For instance, if pl1 < pl2 , firm 1 makes higher profits by increasing pl1 at some node. Hence, let pl denote the infimum of the support of F1 and F2 . Since profits are positive, pl > 0. Suppose that there is an interval (p, p0 ), pl < p < p0 ≤ 1, such that F2 (p) = F2− (p0 ). Without loss of generality, we can assume that F2 (p00 ) < F2 (p) for p00 < p. It follows that F1 (p) = F1− (p0 ) since the profits of firm 1 from any strategy (p00 , x), p00 ∈ (p, p0 ), in the support of its equilibrium strategy are strictly lower than the profits from (p00 +ε, x), where ε > 0 is sufficiently small. We now show that there exists no x ∈ X such that (p, x) is a best-reply for either firm. If neither F1 nor F2 have a mass point at p, then firm i can profitably deviate from any (p − ε, x), where ε > 0 is sufficiently small, to (p00 , x), p00 ∈ (p, p0 ). Suppose then that F2x has a mass point at p for some x ∈ X. Such a mass point is a best-reply for firm 2 only if firm 1 also has a mass point at (p, y) for some y for which π (x, y) > 0 - otherwise, deviating to (p + ε, x) would be profitable for firm 2, for a sufficiently small ε > 0. But then firm 1 can profitably deviate from (p, y) to (p − ε, y) for a sufficiently small ε > 0. This concludes the proof.

6.2

Proposition 2

Define X A = {x ∈ X : π (y, x) = 1 for all y ∈ X}. Suppose that F1 (0) = 1. Then, firm 1 makes zero profits. It follows that F2 (0) = 1 and hence firm 2 also makes zero profits. Obviously, Supp (λi ) ⊆ X A , i = 1, 2, as if λi (x) > 0 and π (y, x) < 1 for some y, firm j can make positive profits charging p = 1 and choosing y. Hence, X A is not empty. Suppose now that X A is not empty. If F1 (0) < 1, then firm 2 makes positive profits. Thus, F2 (0) < 1 and firm 1 also makes positive profits. We first show that it is impossible that π (x, y) = 1 for all x ∈ Supp (λ2 ), y ∈ Supp (λ1 ). Assume the contrary. By Proposition 1, the upper bound of the support of Fi is equal to 1 for i = 1, 2. Take a node z in the support of λ2 such that the upper bound of the support of Fiz is equal to one. The profits of firm 2 are equal to ¢ 1 X¡ 1 − F1x− (1) λ1 (x) 2 x∈X 28

Choosing a price equal to 1 − ε and a node x∗ in X A , firm 2 obtains (1 − ε) X (1 − π (x∗ , x) F1x (1 − ε) + (1 − F1x (1 − ε))) λ1 (x) 2 x∈X Since firm 2’s payoff is positive, F1x− (1) < 1 for some x ∈ Supp (F1 ). But then, for ε sufficiently small, the second expression is larger than the first expression, a contradiction. Now let p∗ be the lowest price p in Supp (F1 ) ∪ Supp (F2 ) for which there exist x ∈ Supp (λj ) and y ∈ Supp (λi ), where i 6= j, such that p ∈ Supp (Fiy ) and π (x, y) < 1. Obviously, p∗ > pl . Without loss of generality, suppose that p∗ ∈ Supp (F2y ). Firm 2’s payoff from the pure strategy (p∗ , y) is ¢ p∗ X ¡ 1 − π (y, x) F1x− (p∗ ) + π (x, y) (1 − F1x (p∗ )) λ1 (x) 2 x∈X If firm 2 deviates to the pure strategy (p∗ − ε, x∗ ), x∗ ∈ X A , it will earn p∗ − ε X (1 − π (x∗ , x) F1x (p∗ − ε) + (1 − F1x (p∗ − ε))) λ1 (x) 2 x∈X By the definition of p∗ , if F1x− (p∗ ) > 0, then π (y, x) = 1. Since π (x, y) < 1 for some x ∈ Supp (λ1 ), for ε sufficiently small, the second expression is larger than the first expression, a contradiction.

6.3

Proposition 6

Consider a bi-symmetric graph (X, π). Define

a = 1 + qY (nY − 1) − qnY b = 1 + qZ (nZ − 1) − qnZ One can verify that weighted regularity holds if and only if the system "

a nY

−b nZ

#"

β1 β2

#

=

"

0 1

#

has a non-negative solution - that is, if and only if ab ≥ 0 (or, equivalently, if and only if (qY∗ − q)(qZ∗ − q) ≥ 0). 29

¡ ¢ Let λ, (F x )x∈Supp(λ) be a symmetric Nash equilibrium strategy, and let F denote the equilibrium marginal pricing strategy. Let S x denote the support of F x , and let pxl and pxu denote the infimum and supremum of S x . Let v x (λ) be the probability that the consumer makes a price comparison conditional on the event that one firm adopts the format x, that is, X λ (y) π (x, y) (7) vx (λ) = y∈X

0

Note that for every x, x0 ∈ Y (similarly, for every x, x0 ∈ Z), vx (λ) = vx (λ) if and only if λ(x) = λ(x0 ). The following claims establish Proposition 6. Lemma 4 F (p) is continuous on [pl , 1]. Proof. It follows from standard arguments, due to the symmetry of equilibrium. Lemma 5 λ (x) = λ (x0 ) for any x, x0 ∈ Y or x, x0 ∈ Z, i = 1, 2. Proof. Suppose that λ (x) > λ (y) for some x, y ∈ Y . Firm i’s payoff from the pure strategy (pxu , x) is  qY λ (y) (1 − F y (pxu )) + , pxu  P P 1 x xu x xu (1 − vx (λ)) x∈Y −(x,y) (1 − F (p )) qY λ (x) + x∈Z (1 − F (p )) qλ (x) + 2 

If the firm deviates to the strategy (pxu , y), it earns

 λ (y) (1 − F y (pxu )) . pxu  P P 1 x xu x xu y (1 − v (1 − F (p )) q λ (x) + (1 − F (p )) qλ (x) + (λ)) Y x∈Y −(x,y) x∈Z 2 

Since λ (x) > λ (y), v (λ) > vy (λ), hence the deviation is profitable. An analogous argument for Z establishes the claim. 0

Lemma 6 For any p ∈ [pl , 1], F x (p) = F x (p) whenever x, x0 ∈ Y or x, x0 ∈ Z. 0

Proof. Suppose that F y (p) > F y (p) for y, y 0 ∈ Y . Firm i’s payoff from the pure strategy (p, y) is  ¢ ¡ y0 (1 − F (p)) λ (y) + qY 1 − F (p) λ (y) +  p P P 1 x x y (1 − F (p)) q λ (x) + (1 − F (p)) qλ (x) + (λ)) (1 − v 0 Y x∈Y −(y,y ) x∈Z 2 

y

30

If the firm deviates to the pure strategy (p, y 0 ), it earns  ¢ ¡ 0 1 − F y (p) λ (y) + qY (1 − F y (p)) λ (y) + p P ¢ . P 1¡ x x y0 1 − v (1 − F (p)) q λ (x) + (1 − F (p)) qλ (x) + (λ) Y x∈Y −(y,y 0 ) x∈Z 2 

0

By Lemma 5, λ(y) = λ(y 0 ) and therefore vy (λ) = v y (λ). It follows that the deviation is profitable. Lemma 7 If λ (x) = 0 for some x ∈ X, then λ verifies weighted regularity. Proof. Suppose that λ(x) = 0 for some x ∈ Y . By Lemma 5, λ is a uniform distribution over Z - thus, in particular, λ(y) = 0 for all y ∈ Y . Therefore, vz (λ) = qZ∗ for every z ∈ Z and vy (λ) = q for every y ∈ Y . If qZ∗ 6= q, it must be profitable to deviate either to the pure strategy (1, y) or to the pure strategy (pl , y). If qZ∗ = q, then λ verifies weighted regularity. Lemma 8 Suppose that λ (x) > 0 for all x ∈ X. Then: (i) If (X, π) is not weighted-regular, either pyu = pzl or pzu = pyl for any y ∈ Y and z ∈ Z. (ii) If pyu = pzl or pzu = pyl for any y ∈ Y and z ∈ Z, (X, π) is not weighted-regular. Proof. (i) Suppose that (X, π) is not weighted-regular and vz (λ) < vy (λ). By Lemma 6, the nodes in Y have the same F y and the nodes in Z have the same F z . Therefore, S y ∩ S z 6= ∅, for any y ∈ Y and z ∈ Z. The following equations must hold in equilibrium. 1 (1 − vy (λ)) = 2 1 λ (z) (1 + qZ (nZ − 1)) (1 − F z (pyu )) + (1 − vz (λ)) 2 ¡¡ ¡ ¢¢¢ 1 λ (z) qnZ + (1 + qY (nY − 1)) λ (y) 1 − F y pzl + (1 − vy (λ)) = 2 ¡¡ ¡ ¢¢¢ 1 + (1 − v z (λ)) λ (z) (1 + qZ (nZ − 1)) + qnY λ (y) 1 − F y pzl 2 λ (z) qnZ (1 − F z (pyu )) +

which simplify to ¡ ¡ ¢¢ v z (λ) − vy (λ) bλ (z) (1 − F z (pyu )) = bλ (z) − aλ (y) 1 − F y pzl = 2 31

Hence, b < 0. Since the graph is not weighted regular, a > 0. It can be easily verified ¡ ¢ that the above equations hold only if F z (pyu ) = 0 and F y pzl = 1. If vz (λ) > vy (λ), a symmetric argument establishes the claim. (ii) Suppose that pyu = pzl . Note that v z (λ) − vy (λ) = bλ (z) − aλ (y) In equilibrium bλ (z) − aλ (y) 2 Since λ (y) , λ (z) > 0, we have ab < 0. A symmetric argument establishes the claim for the case pzu = pyl . bλ (z) =

Lemma 9 Suppose that λ (x) > 0 for any x ∈ X. If pyu 6= pzl and pzu 6= pyl for any y ∈ Y and z ∈ Z, then λ verifies weighted regularity, max-min payoffs are obtained, and F z (p) = F y (p) for any p ∈ [pl , 1]. Proof. Lemma 8 implies that if pyu 6= pzl and pzu 6= pyl for any y ∈ Y and z ∈ Z then the graph is weighted-regular. As in the proof of Lemma 8, the following equilibrium conditions must hold bλ (z) (1 − F z (pyu )) =

bλ (z) − aλ (y) 2

¡ ¢¢ bλ (z) − aλ (y) ¡ bλ (z) − aλ (y) 1 − F y pzl = 2 First note that if either b = 0 or a = 0, then either λ (y) = 0 or λ (z) = 0. Hence, ¡ ¡ ¢¢ suppose that ab > 0. Setting (1 − F z (pyu )) = σ and 1 − F y pzl = δ, rewrite the system in matrix notation as  " # " # a b bσ − λ (z) 0   2 2 =   b a λ (y) 0 −aδ + 2 2 

This system has a non-null solution if and only if −σ − δ + 2σδ + 1 = 0 which is only possible, for 0 ≤ δ, σ ≤ 1, when δ = 1, σ = 0 or δ = 0, σ = 1. In the 32

former case, viz (λ) = viy (λ) and thus λ verifies weighted regularity. In the latter case, bλ (z) =

bλ (z) − aλ (y) 2

and hence positive solutions for λ (z) , λ (y) do not exist when ab > 0. Thus in equilib¡ ¢ rium, F z (pyu ) = 1, F y pzl = 0, and v z (λ) = vy (λ). Consequently, for any p ∈ [pl , 1] bλ (z) (1 − F z (p)) = aλ (y) (1 − F y (p))

Since v z (λ) − vy (λ) = bλ (z) − aλ (y) = 0, we have F z (p) = F y (p). Part (i) of the proposition follows from Lemmas 7, 8, and 9. If qY∗ < q < qZ∗ , then a symmetric Nash equilibrium must be a cutoff equilibrium by Lemmas 7 and 8. Moreover, by Lemma 6, it suffices to consider two cases: either λU is a uniform distribution over Y and λL is a uniform distribution over Z, or λU is a uniform distribution over Z and λL is a uniform distribution over Y . To pin down the framing strategy λ, we use the equilibrium condition that firms are indifferent between playing y ∈ Y and z ∈ Z at the cutoff price pm (pm = pzu = pyl in the former case, and pm = pzl = pyu in the latter case). In the former case, the condition is given by the equation λ (y) nY q − λ (z) nZ qZ∗ = λ (y) nY qY∗ − λ (z) nZ q for arbitrary y ∈ Y and z ∈ Z. In the latter case, the condition is given by the equation λ (z) nZ q − λ (y) nY qY∗ = λ (z) nZ qZ∗ − λ (y) nY q for arbitrary y ∈ Y and z ∈ Z. Since qY∗ < q < qZ∗ , the latter case is ruled out, and the former equation yields λ.

6.4

Proposition 7

¡ ¢ Let λi , (Fix )x∈Supp(λi ) i=1,2 be a Nash equilibrium. Given a framing strategy λ and a node x ∈ X, define v x (λ) as in (7). Note that in a symmetric graph, vx (λ) = v y (λ) if and only if λ(x) = λ(y). The proof follows from the following claims.

Lemma 10 Fi (p) is continuous on [pl , 1), i = 1, 2.

33

Proof. Consider first the case q = 0. Suppose that Fjx has a mass point at p ∈ [pl , 1). Firm i’s payoff from the pure strategy (p, x) is ¡ ¢ ¢ 1¡ x 1 p · [ 1 − Fjx (p) λj (x) + Fj (p) − Fjx− (p) λj (x) + (1 − λj (x))] 2 2

The firm’s payoff from (p + ε, x) is bounded from above by

¡ ¢ 1 (p + ε) · [ 1 − Fjx (p) λj (x) + (1 − λj (x))] 2

and the firm’s payoff from (p − ε, x) is bounded from below by (p − ε) · [(1 − Fjx− (p))λj (x) +

1 (1 − λj (x))] 2

If ε > 0 is sufficiently small, the third expression is strictly larger than the first two. Since the second expression is increasing in ε, it follows that for small ε, Fix is constant on the interval (p, p+ε). But then firm j can profitably deviate from (p, x) to (p+ 2ε , x). The proof for the case q > 0 is more conventional. If Fjx has a mass point at some p ∈ [pl , 1), then Fi must be constant on an interval [p, p + ε) to be a best-reply, contradicting Proposition 1. Lemma 11 λi (x) =

1 for all x ∈ X, i = 1, 2. n

xu Proof. Let Ximax = arg maxx∈X λi (x). Let pxl i and pi be the infimum and supremum l l of the support of Fix , and let Xil be the set of nodes such that pxl i = p any x ∈ Xi . By Proposition 1, Xil is non-empty and is a subset of Xjmax , i 6= j. Suppose that 1 maxx∈X λ1 (x) > . We consider two cases. n

Case 1: Xil = Xjl . We first show that Xil = Xjmax , i 6= j. Suppose not and let x0 ∈ Xil and x00 ∈ Xjmax − Xil . By hypothesis, x00 ∈ / Xjl . For any sufficiently small ε > 0, firm i’s payoff from the pure strategy (pl + ε, x0 ) is 

 ¡ ¢¢ ¡ l x0 0 00 1 − Fj p + ε λj (x ) + qλj (x ) + ¡ l ¢ p +ε  P ¡ ¡ l ¢¢ ¢  1¡ x x0 1 − v 1 − F p + ε qλ (x) + (λ ) 0 00 j j i j x∈X−(x ,x ) 2

If the firm deviates to (pl + ε, x00 ), it earns 

 ¡ ¢¢ ¡ l x0 0 λj (x ) + 1 − Fj p + ε qλj (x ) + ¢ ¡ l p +ε  P ¢¢ ¢  ¡ ¡ l 1¡ x x00 + ε qλ (x) + (λ ) 1 − v 1 − F p 0 00 j j j i x∈X−(x ,x ) 2 00

34

0

00

Since x0 , x00 ∈ Xjmax , λj (x0 ) = λj (x00 ) and hence vix (λj ) = vix (λj ). Since by hypothesis ¢¢ 0 ¡¡ Fjx pl + ε > 0, firm i’s deviation is profitable. 1 If maxx∈X λ2 (x) = , then, X2max = X and, since X1max = X2max , maxx∈X λ1 (x) = n 1 1 , which is false by hypothesis. Hence, maxx∈X λ2 (x) > . It follows that pxu i < 1 for n n l x0 u xu l any x ∈ Xi , i = 1, 2. Suppose that p1 is the highest pi for any x ∈ Xi , i = 1, 2. Let 0 / X2max . Firm 1’s payoff from the pure strategy (p1x u , x0 ) is x00 be such that x00 ∈  ¡ ¡ x0 u ¢¢ 00 x00 (x ) 1 − F p + qλ j j 1 0 p1x u  P ¢¢ ¢  ¡ ¡ l 1¡ x x0 1 − v + ε qλ (x) + (λ ) 1 − F p j j i j x∈X−(x0 ,x00 ) 2 

0

If the firm deviates to (p1x u , x00 ), it earns

 ¡ ¡ x0 u ¢¢ 00 x00 (x ) 1 − F p + λ j 1 j 0 p1x u  P ¢¢ ¢  ¡ ¡ l 1¡ x x00 1 − v + ε qλ (x) + (λ ) 1 − F p j j j i x∈X−(x0 ,x00 ) 2 

0

00

Since vix (λj ) > vix (λj ), this deviation is profitable.

Case 2: Xil 6= Xjl . We first show that Xil ∩ Xjl = ∅. Suppose not and let x0 ∈ Xil ∩ Xjl and x00 ∈ Xil − Xjl . For any sufficiently small ε > 0, firm i’s payoff from the pure strategy (pl + ε, x0 ) is 

 ¡ ¢¢ ¡ l x0 0 00 1 − F + ε λ (x ) + qλ (x ) + p ¡ l ¢ j j j p +ε  P ¢¢ ¢  ¡ ¡ 1¡ x l x0 1 − v + ε qλ (x) + (λ ) 1 − F p 0 00 j j j i x∈X−(x ,x ) 2

If the firm deviates to (pl + ε, x00 ), it earns 

 ¡ ¢¢ ¡ l 00 x0 0 λ (x ) + 1 − F + ε qλ (x ) + p ¡ l ¢ j j j p +ε  P ¢¢ ¢  ¡ ¡ 1¡ 00 x l 1 − vix (λj ) qλj (x) + x∈X−(x0 ,x00 ) 1 − Fj p + ε 2 0

00

Since x0 , x00 ∈ Xjmax , λj (x0 ) = λj (x00 ) and hence vix (λj ) = vix (λj ). As by hypothesis ¢¢ 0 ¡¡ Fjx pl + ε > 0, firm i’s deviation is profitable. xl ¯ i = {x ∈ X | x ∈ / Xjmax or pxl ¯i = minx∈X¯i pxl Let X i ≥ pj , i 6= j}. Let p i . Since Xil ∩ Xjl = ∅, p¯i > pl . Also, mini p¯i < 1. To see this, note that, for q > 0, only one firm can have a mass point at p = 1 (by standard arguments) and, if Fix has no mass xl x point at x ∈ Xjl ⊂ Ximax , pxl j ≤ pi < 1. For q = 0, if Fi (p) = 0 for any p < 1 at a node x ∈ Xjl ⊂ Ximax , then firm j makes higher profits raising pl at node x. 35

0

Suppose that p¯i ≤ p¯j . We first show that it cannot be the case that p¯i = pix l for / Xjmax . Let x00 be a node in Xjmax ∩ Xil . Firm i’s payoff from the pure strategy x0 ∈ (¯ pi , x0 ) is  ¡ ¢ 0 pi ) + qλj (x00 ) + λj (x0 ) 1 − Fjx (¯ p¯i  P ¢ ¢  ¡ 1¡ x x0 1 − v (¯ p ) qλ (x) + (λ ) 1 − F 0 00 i j j i j x∈X−(x ,x ) 2 

If the firm deviates to (¯ pi , x00 ), it earns

 ¡ ¢ 0 pi ) + λj (x00 ) + qλj (x0 ) 1 − Fjx (¯ p¯i  P ¢ ¢  ¡ 1¡ x x00 (¯ p ) qλ (x) + (λ ) 1 − v 1 − F i j j i j x∈X−(x0 ,x00 ) 2 

00

0

Since vix (λj )−vix (λj ) = (1 − q) (λj (x00 ) − λj (x0 )), it can be verified that the deviation is profitable. 0 0 Suppose that p¯i = pix l for x0 ∈ Xjmax . If pjx l < p¯i , a contradiction is easily obtained pi , x) is a profitable deviation (recall that p¯i ≤ p¯j as for any x ∈ Xil , the strategy (¯ x0 l by hypothesis). If pj = p¯i , then p¯i = p¯j and, by the same argument, Fjx (p) = 0 for ¯j = pxl any x ∈ Xil in an interval [pl , p¯i + ε) for some ε > 0 (if not, pxl i < p j for some l pj , x˜) for x˜ ∈ Xjl ). Hence, node x in Xi and firm j would obtain higher profits with (¯ / Xil . Let x00 ∈ Xil . Then, for any γ ∈ (0, ε), firm i’s payoff from the pure strategy x0 ∈ 0 (pix l + γ, x0 ) is  ¡ ¢¢ 0 ¡ 0 λj (x0 ) 1 − Fjx pix l + γ + qλj (x00 ) + ¢¢ ¢  ¡ ¡ x0 l 1¡ x x0 1 − v + γ qλ (x) + (λ ) 1 − F p j j i j i x∈X−(x0 ,x00 ) 2

 ³ 0 ´ pxi l + γ  P

0

If the firm deviates to (pix l + γ, x00 ), it earns

 ¡ ¢¢ 0 ¡ 0 λj (x00 ) + qλj (x0 ) 1 − Fjx pix l + γ + ¢¢ ¢  ¡ ¡ x0 l 1¡ x x00 1 − v + γ qλ (x) + (λ ) 1 − F p j j j i i x∈X−(x0 ,x00 ) 2 ¢ 0 00 0 ¡ 0 Since λj (x0 ) = λj (x00 ), vix (λj ) = vix (λj ) and Fjx pix l + γ > 0, one can verify that the deviation is profitable.  ³ 0 ´ pix l + γ  P

Lemma 12 F1 and F2 have no mass points.

36

Proof. By Lemma 10, F1 and F2 have no mass points at any p < 1. Firm i’s profits from the pure strategy (1, x) are 1 1 1 X · (1 − Fjx− (1) − qFiy− (1)) 2 n n y∈X−x Note that (1, x) is optimal only if Fjx− (1) ≤ Fiy− (1) for any y ∈ X. Hence, if both F1 and F2 have mass points at p = 1, there must exist a node x such that F1x and F2x have mass points at p = 1. By standard arguments, a contradiction is obtained. Thus, suppose that F2 has no mass point at p = 1. By Lemma 11, firm 1’s payoff is then 1 (1 − q) (n − 1) · 2 n If firm 2 plays the pure strategy (1 − ε, x), it earns at least (1 − ε)

1 (1 − ε) (1 − q) (n − 1) (1 − F1x (1 − ε)) + n 2 n

Since, limε−→0 F1x (1 − ε) < 1, firm 20 s payoff is strictly above firm 1’s payoff. This implies that the supports of F1 and F2 have different infima, contradicting Proposition 1. 0

Lemma 13 Fix (p) = Fjx (p) for any p ∈ [pl , 1], x, x0 ∈ X, i, j ∈ {1, 2}. Proof. We first show that Fix (p) must be strictly increasing for any x ∈ X and i = 1, 2. Suppose not. Since Fi is strictly increasing, i = 1, 2, there exist a firm k, 0 0 y, y 0 ∈ X, and p0 < p00 such that Fky (p) > Fky (p) for p ∈ [p0 , p00 ), and Fix (p) = Fjx (p) for any p ∈ [p00 , 1], x, x0 ∈ X, i, j ∈ {1, 2}. For simplicity, assume k = 1. Then, firm 2’s payoff from the pure strategy (p, y), p ∈ [p0 , p00 ), is ³ ´  y y0 (p)) + q 1 − F (p) + (1 − F 1 1 p   · P (1 − q) (n − 1) x n x∈X−(y,y 0 ) q (1 − F1 (p)) + 2 

If the firm deviates to (p, y 0 ), it earns ´ ³  y0 y (p) + q (1 − F (p)) + 1 − F 1 1 p   · P (1 − q) (n − 1) x n x∈X−(y,y 0 ) q (1 − F1 (p)) + 2 

37

0

Since F1y (p) > F1y (p), the deviation is profitable. Hence, F2y (p) is constant on [p0 , p00 ). 00 Since F2 is strictly increasing, there exists a node y 00 such that F2y (p) > F2y (p) for p ∈ [p0 , p00 ). It follows that F1y is also constant on [p0 , p00 ). Now let pˆ be lowest price p0 for 00 0 p) > F2y (ˆ p) and F1y (ˆ p) > F1y (ˆ p), which Fiy is constant on [p0 , p00 ) for i = 1, 2. Since F2y (ˆ a contradiction is obtained showing, by analogous arguments, the strategy (ˆ p, y) is not a best-reply. Since every Fix (p) is strictly increasing for any x ∈ X and i = 1, 2, it can easily be verified that its value is determined by a system of linear equations which has a unique, symmetric solution.

6.5

Proposition 9

(i) Whenever p1 ≤ p2 , the consumer chooses firm 1 with probability one. Whenever p1 > p2 , the consumer chooses firm 2 if and only if he makes a price comparison. Therefore, for every price p that lies strictly above the infimum of Supp(F2 ), firm 1’s optimal format minimizes v(·, λL2 (p)), where λL2 (p) denotes firm 2’s framing strategy conditional on p0 < p. Similarly, for every price p that lies strictly below the supremum of Supp(F1 ), firm 2’s optimal format maximizes v(λU1 (p), ·), where λU1 (p) denotes firm 1’s framing strategy conditional on p0 > p. It can be verified that Proposition 1 extends to the Incumbent-Entrant model. Therefore, Supp(F1 ) and Supp(F2 ) have the same infimum pl < 1 and the same supremum pu = 1. Therefore, in Nash equilibrium, firm 1’s framing strategy conditional on p > pl and firm 2’s framing strategy conditional on p < 1 constitute a Nash equilibrium in the associated hide-and-seek game. These framing strategies are equal to the firms’ marginal equilibrium framing strategies, because as we will verify below, F1 does not have an atom on pl and F2 does not have an atom on p = 1. (ii) Since p = 1 is in the support of F1 and firm 2’s framing strategy conditional on p < 1 max-minimizes v, firm 1’s equilibrium payoff is 1 − v ∗ . Since firm 1 is chosen with probability one when it charges pl , it follows that pl = 1 − v ∗ . But since firm 1’s framing strategy conditional on p > pl min-maximizes v, it follows that firm 2’s payoff is v ∗ · (1 − v ∗ ). (iii) The formulas of F1 and F2 follow directly from the condition that every p ∈ (1 − v∗ , 1) maximizes each firm’s profit given the opponent’s strategy, and the characterization of firm 1’s framing strategy conditional on p > pl and firm 2’s framing strategy conditional on p < 1.

38

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